Question 1

Does the point (-5, -1, -5)lie on any of the level surfaces of $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 }$ ? If so, what is the equation of that level surface?

Accepted Answer

A) True 
 B)False

Question 2

Level surfaces of g(x,y,z)corresponding to different levels cannot intersect.

Accepted Answer

A) True 
 B)False

Question 3

The points A = (1, 1, 1), B = (2, 4, 2), and C = (3, 2, 2)are the vertices of a triangle in space. Which of the vertices is closest to the origin?

Accepted Answer

A) A 
B) B 
C) C

Question 4

If all of the contours of a function f(x, y)are parallel lines, then the function must be linear.

Accepted Answer

A) True 
 B)False

Question 5

The level surfaces of the functions $$g _ { 1 } ( x , y , z )$$ $$g _ { 2 } ( x , y , z )$$ and $$g _ { 3 } ( x , y , z )$$ are shown in below respectively.    Which function is is decreasing in the positive z-direction?

Accepted Answer

The answer of The level surfaces of the functions \[g...

Question 6

Let f(x, y, z)= ax + by + cz + d be a linear function of three variables, for constants a, b, c and d.
Given that some of the cross sections of f are f(x, 1, 1)= 0 + 4x, f(0, y, -3)= 4 + 4y,
and f(1, 1, z)= 6 - 2z, find a formula for f.

Accepted Answer

The answer of Let f(x, y, z)= ax + by...

Question 7

Match the graph with the function.

Accepted Answer

A)  $f ( x , y ) = 6 - 2 x + 3 y$ 
B)  $f ( x , y ) = 6 - 2 x - 3 y$ 
C)  $f ( x , y ) = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$ 
D)  $f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$

Question 8

A certain piece of electronic surveying equipment is designed to operate in temperatures ranging from 0° C to 30° C.Its performance index, p(t, h), measured on a scale from 0 to 1, depends on both the temperature t and the humidity h of its surrounding environment.Values of the function p = f(t, h)are given in the following table.(The higher the value of p, the better the performance.)  
What is the value of p(0, 25)?

Accepted Answer

The answer of A certain piece of electronic surveying equipment...

Question 9

Sketch the level set of $f ( x , y ) = x ^ { 2 } + z ^ { 2 }$ corresponding to f = 16 and describe it in words.

Accepted Answer

The answer of Sketch the level set of \(f (...

Question 10

Yummy Potato Chip Company has manufacturing plants in N.Y.and N.J.The cost of manufacturing depends on the quantities (in thousand of bags), q₁ and q₂, produced in the N.Y.and N.J.factories respectively.Suppose the cost function is given by $C \left( q _ { 1 } , q _ { 2 } \right) = 2 q _ { 1 } ^ { 2 } + q _ { 1 } q _ { 2 } + q _ { 2 } ^ { 2 } + 420$ (a)Find $C ( 10,25 )$ (b)By comparing the terms $2 q _ { 1 } ^ { 2 }$ and $q _ { 2 } ^ { 2 }$ in the above expression, the manager concluded that it is more expensive to produce in the N.Y.factory.Will shifting all the production to the N.J.factory minimize the production cost?

Accepted Answer

The answer of Yummy Potato Chip Company has manufacturing plants...

Question 11

The picture below is the contour diagram of f(x, y).The areas between contours have been shaded.Lighter shades represent higher levels, while darker shades represent lower levels.  Sketch the cross section of f(x, y)with x fixed at x = 0.5.

Accepted Answer

The answer of The picture below is the contour diagram...

Question 12

Suppose f is a function of two variables. Given a point (x,y)in the domain of f, there is a level curve of f passing through (x,y).

Accepted Answer

A) True 
 B)False

Question 13

A certain piece of electronic surveying equipment is designed to operate in temperatures ranging from 0° C to 30° C.Its performance index, p(t, h), measured on a scale from 0 to 1, depends on both the temperature t and the humidity h of its surrounding environment.Values of the function p = f(t, h)are given in the following table.(The higher the value of p, the better the performance.)  Describe the function p(10, h)and explain its meaning.

Accepted Answer

The answer of A certain piece of electronic surveying equipment...

Question 14

Given the table of some values of a linear function determine a formula for the function.

Accepted Answer

The answer of Given the table of some values of...

Question 15

Match the following function with the graphs below. The function z = f(x, y)giving happiness as a function of health y and money x according to the statement of a fortune cookie: 'Whoever said money cannot buy happiness does not know where to shop.'

Accepted Answer

A)   
B)   
C)

Question 16

Suppose f is a function of two variables. The cross-sections of f with one variable fixed must also be functions.

Accepted Answer

A) True 
 B)False

Question 17

Match the appropriate function $$f ( x , y , z )$$ with the geometric description of its level surface f = 0.

Accepted Answer

The answer of Match the appropriate function \[f ( x...

Question 18

Find a linear function with f(-5, 5)= 1, whose graph is parallel to the level surfaces of g(x, y, z)= 10x + 15y + 5z.

Accepted Answer

The answer of Find a linear function with f(-5, 5)=...

Question 19

The points A = (4, 1, 2), B = (3, -2, 3), and C = (-2, 3, -4)are the vertices of a triangle in space. Which of the vertices is closest to the yz-plane?

Accepted Answer

A) C 
B) A 
C) B

Question 20

Consider the plane that passes through the points (1, 3, 10), (-1, -1, 0), and (2, 1, -1).
If you were walking on this plane with no change in altitude, what would be the slope of your path in the xy-plane?

Accepted Answer

The answer of Consider the plane that passes through the...

Exam 12: Functions of Several Variables

Does the point (-5, -1, -5)lie on any of the level surfaces of $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 }$ ? If so, what is the equation of that level surface?

Level surfaces of g(x,y,z)corresponding to different levels cannot intersect.

The points A = (1, 1, 1) , B = (2, 4, 2) , and C = (3, 2, 2) are the vertices of a triangle in space. Which of the vertices is closest to the origin?

If all of the contours of a function f(x, y)are parallel lines, then the function must be linear.

Let f(x, y, z)= ax + by + cz + d be a linear function of three variables, for constants a, b, c and d. Given that some of the cross sections of f are f(x, 1, 1)= 0 + 4x, f(0, y, -3)= 4 + 4y, and f(1, 1, z)= 6 - 2z, find a formula for f.

Match the graph with the function. $Match the graph with the function. A) f ( x , y ) = 6 - 2 x + 3 y B) f ( x , y ) = 6 - 2 x - 3 y C) f ( x , y ) = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } D) f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$

Sketch the level set of $f ( x , y ) = x ^ { 2 } + z ^ { 2 }$ corresponding to f = 16 and describe it in words.

The picture below is the contour diagram of f(x, y).The areas between contours have been shaded.Lighter shades represent higher levels, while darker shades represent lower levels. Sketch the cross section of f(x, y)with x fixed at x = 0.5.

Suppose f is a function of two variables. Given a point (x,y)in the domain of f, there is a level curve of f passing through (x,y).

Given the table of some values of a linear function determine a formula for the function.

Match the following function with the graphs below. The function z = f(x, y) giving happiness as a function of health y and money x according to the statement of a fortune cookie: 'Whoever said money cannot buy happiness does not know where to shop.'

Suppose f is a function of two variables. The cross-sections of f with one variable fixed must also be functions.

Match the appropriate function $f ( x , y , z )$ with the geometric description of its level surface f = 0.

Find a linear function with f(-5, 5)= 1, whose graph is parallel to the level surfaces of g(x, y, z)= 10x + 15y + 5z.

The points A = (4, 1, 2) , B = (3, -2, 3) , and C = (-2, 3, -4) are the vertices of a triangle in space. Which of the vertices is closest to the yz-plane?

Consider the plane that passes through the points (1, 3, 10), (-1, -1, 0), and (2, 1, -1). If you were walking on this plane with no change in altitude, what would be the slope of your path in the xy-plane?

Exam 12: Functions of Several Variables

Does the point (-5, -1, -5)lie on any of the level surfaces of f(x,y,z)=x2+4y2+z2f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 }f(x,y,z)=x2+4y2+z2 ? If so, what is the equation of that level surface?

Level surfaces of g(x,y,z)corresponding to different levels cannot intersect.

The points A = (1, 1, 1) , B = (2, 4, 2) , and C = (3, 2, 2) are the vertices of a triangle in space. Which of the vertices is closest to the origin?

If all of the contours of a function f(x, y)are parallel lines, then the function must be linear.

The level surfaces of the functions g1(x,y,z)g _ { 1 } ( x , y , z )g1​(x,y,z) g2(x,y,z)g _ { 2 } ( x , y , z )g2​(x,y,z) and g3(x,y,z)g _ { 3 } ( x , y , z )g3​(x,y,z) are shown in below respectively. Which function is is decreasing in the positive z-direction?

Let f(x, y, z)= ax + by + cz + d be a linear function of three variables, for constants a, b, c and d. Given that some of the cross sections of f are f(x, 1, 1)= 0 + 4x, f(0, y, -3)= 4 + 4y, and f(1, 1, z)= 6 - 2z, find a formula for f.

Match the graph with the function.

Sketch the level set of f(x,y)=x2+z2f ( x , y ) = x ^ { 2 } + z ^ { 2 }f(x,y)=x2+z2 corresponding to f = 16 and describe it in words.

The picture below is the contour diagram of f(x, y).The areas between contours have been shaded.Lighter shades represent higher levels, while darker shades represent lower levels. Sketch the cross section of f(x, y)with x fixed at x = 0.5.

Suppose f is a function of two variables. Given a point (x,y)in the domain of f, there is a level curve of f passing through (x,y).

Given the table of some values of a linear function determine a formula for the function.

Match the following function with the graphs below. The function z = f(x, y) giving happiness as a function of health y and money x according to the statement of a fortune cookie: 'Whoever said money cannot buy happiness does not know where to shop.'

Suppose f is a function of two variables. The cross-sections of f with one variable fixed must also be functions.

Match the appropriate function f(x,y,z)f ( x , y , z )f(x,y,z) with the geometric description of its level surface f = 0.

Find a linear function with f(-5, 5)= 1, whose graph is parallel to the level surfaces of g(x, y, z)= 10x + 15y + 5z.

The points A = (4, 1, 2) , B = (3, -2, 3) , and C = (-2, 3, -4) are the vertices of a triangle in space. Which of the vertices is closest to the yz-plane?

Consider the plane that passes through the points (1, 3, 10), (-1, -1, 0), and (2, 1, -1). If you were walking on this plane with no change in altitude, what would be the slope of your path in the xy-plane?

Does the point (-5, -1, -5)lie on any of the level surfaces of $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 }$ ? If so, what is the equation of that level surface?

Match the graph with the function. $Match the graph with the function. A) f ( x , y ) = 6 - 2 x + 3 y B) f ( x , y ) = 6 - 2 x - 3 y C) f ( x , y ) = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } D) f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$

Sketch the level set of $f ( x , y ) = x ^ { 2 } + z ^ { 2 }$ corresponding to f = 16 and describe it in words.

Match the appropriate function $f ( x , y , z )$ with the geometric description of its level surface f = 0.